Topics and Objectives
What is covered, and what you should be able to do with it
The list below is the plan for the term. Topics and their order may change.
| Weeks | Phase | Primary deliverable |
|---|---|---|
| 1 | Orientation; physics and units | Practice problem |
| 2–6 | Foundation | Eight problems weekly, solved and repaired; 3 problem sets |
| 7–9 | Knowledge maps: constituents of the universe | One map per session, two a week |
| 10–11 | Project clinic and prototype, with the TA | Research question + prototype |
| 12–13 | Knowledge maps: current and future probes | One map per session, two a week |
| 14–15 | Capstone symposium | Research talk |
Week 1: orientation
I take both meetings this week. It is the only week I do.
Review of physics: elementary particles and their fundamental interactions. Unit systems: natural, Planck and astronomical units. Index notation. Basic observational properties of our universe.
The conventions fixed here are used for fifteen weeks, and sign errors later in the term are traced back to this list.
Objectives
- Students can define the natural, Planck and astronomical unit systems, explain the rationale for each, and convert quantities among them and to conventional units.
- Students can distinguish the Planck mass from the reduced Planck mass and apply each in its conventional context.
- Students can describe the particle content of the standard model and the role of each species in the visible universe.
- Students can explain the four fundamental interactions and the scales on which each is relevant.
- Students can apply index notation and the Einstein summation convention to manipulate tensor expressions.
- Students can state the metric signature and curvature sign conventions used in this course and reconcile a result obtained under a different convention.
Weeks 2–6: Foundation
One topic per week across both meetings, eight problems a week, four per meeting.
| Week | Topic | |
|---|---|---|
| 2 | FRW geometry and kinematics | Hubble’s law and its uniqueness; comoving and physical distance; the general form of the metric; homogeneous isotropic 2-spaces; the three 3-spaces; cosmological redshift; why a massive particle’s velocity decays; why every species is relativistic early |
| 3 | Friedmann equations and expansion history | Why the contents must be a perfect fluid; the Einstein equation for FRW; energy-momentum conservation and its redundancy; the heuristic derivation and why it fails; density parameters and the acceleration condition; matter, radiation and topological defects; the static universe and its instability; integrating \(\Lambda\)CDM |
| 4 | Age, horizons, and distances | The age integral; age and acceleration; particle and comoving horizons; a universe with no beginning; comoving distance; angular diameter distance and its turnover; luminosity distance; distance duality |
| 5 | Thermodynamics in the expanding universe | Phase space in FRW and why \(f\) is a scalar; cosmological dimming from Liouville; \(n\), \(\rho\), \(P\) and \(s\) from the distribution function; the two limits; the end of the antiparticle background; effective relativistic degrees of freedom; the neutrino temperature and the age of the universe |
| 6 | Inflation | The flatness problem; the horizon problem; how much inflation; what inflation does not do; the inflaton; slow roll; two potentials; slow roll checked numerically |
Week 2: FRW geometry and kinematics
Objectives
- Students can state Hubble’s law and prove that it is the unique expansion law consistent with homogeneity and isotropy, identifying which step requires which assumption.
- Students can distinguish comoving from physical distance and explain why a recession velocity exceeding \(c\) violates nothing.
- Students can derive the general form of the FRW metric from spatial symmetry and distinguish its physical content from its coordinate choices.
- Students can construct the metric of a homogeneous isotropic three-space of any curvature and state on what the claim that there are exactly three rests.
- Students can derive the cosmological redshift from the geodesic equation and obtain the same result independently from the properties of comoving observers.
- Students can show that every particle species behaves as radiation at sufficiently early times.
Week 3: Friedmann equations and expansion history
Objectives
- Students can show that homogeneity and isotropy force the energy-momentum tensor into perfect fluid form, and identify what is thereby excluded.
- Students can obtain the Friedmann equations from the Einstein equation and verify the result by independent symbolic computation and by physical limits.
- Students can explain why the continuity equation is not independent of the Friedmann equations.
- Students can critique the Newtonian and thermodynamic derivations of the Friedmann equations and separate the flaws that are fatal from those that are repairable.
- Students can solve the continuity equation for a component of given equation of state and derive, rather than quote, the equation of state of matter, radiation and topological defects.
- Students can determine the condition for accelerated expansion and identify which components satisfy it.
- Students can integrate the Friedmann equation numerically for a given cosmology and establish that the result is correct.
Week 4: Age, horizons and distances
Objectives
- Students can compute the age of the universe and the look-back time, and state what the age must be compared against for the comparison to constrain anything.
- Students can relate the age of the universe to the sign of the acceleration, and state the hypothesis that relation requires.
- Students can derive the particle and comoving horizons and state the conditions under which each exists.
- Students can define the comoving, angular diameter and luminosity distances, compute each for a given cosmology, and account for the turnover in the angular diameter distance.
- Students can prove the distance duality relation and identify what it assumes and what it does not.
Week 5: Thermodynamics in the expanding universe
Objectives
- Students can construct the phase space of the FRW universe and explain why the distribution function is a scalar.
- Students can derive the number density, energy density, pressure and entropy density of a species from its distribution function.
- Students can evaluate these in the relativistic and non-relativistic limits and account for the \(7/8\) and \(3/4\) factors separately.
- Students can determine when a particle-antiparticle background disappears and identify what sets the answer.
- Students can define the effective relativistic degrees of freedom for energy, entropy and number, count them for the standard model, and explain why three separate counts are needed.
- Students can derive the photon-to-neutrino temperature ratio from comoving entropy conservation and identify the ordering of events on which it depends.
- Students can relate temperature, redshift and cosmic time, and assess the accuracy of the constant-\(g_\star\) approximation.
Week 6: Inflation
Objectives
- Students can state the flatness and horizon problems quantitatively and demonstrate that both require an early accelerated epoch.
- Students can estimate the required number of e-folds and identify what that number depends on.
- Students can evaluate the claim that inflation explains the homogeneity of the universe.
- Students can derive the equation of motion of a homogeneous scalar field and show that acceleration requires the potential to dominate the kinetic energy.
- Students can derive the slow-roll parameters and state which physical requirement produces each.
- Students can compute the slow-roll parameters and the number of e-folds for a given potential, and verify the slow-roll approximation numerically.
Two sets of topics are deliberately absent. The classical cosmological tests are applications of the week 4 machinery rather than foundations of it, and are set as a problem set. Everything downstream of equilibrium thermodynamics enters in the map phase as literature rather than as derivation.
Weeks 7–9 and 12–13: Knowledge maps
Ten sessions, ten subjects, one subject per session. A bounded paper set for each. The goal is not paper summaries; it is the structure of the subject.
Constituents of the universe, in order of how well we know them.
| 1 | Photons | The spectrum, recombination, and the polarized sky |
| 2 | Baryons | Two routes to \(\Omega_b\); the lithium problem; the cold gas; reionization |
| 3 | Neutrinos | How many; the mass from oscillations, from tritium, and from structure |
| 4 | Dark matter | Cold or warm; the WIMP; how light it can be; whether it interacts with itself |
| 5 | Dark energy | Geometry from supernovae and BAO; dynamics from RSD and ISW |
| 6 | Initial conditions | Primordial spectra, the transfer function, \(P(k)\), non-Gaussianity |
Current and future probes.
| 7 | DESI, HETDEX | Distances and clustering from spectroscopy |
| 8 | Roman, Euclid, LSST | Shapes from imaging: three designs for one measurement |
| 9 | LVK, LISA, PTA | Strain: sirens without a ladder, and backgrounds as a thermometer |
| 10 | SPHEREx | The statistics of the field itself |
Each constituent session maps four things we still want to know about that subject. Each probe session follows an instrument from what it records to what it can settle. See the syllabus for how these work.
Weeks 10–11: Project clinic and prototype
The instructor is away. Meetings continue on the normal schedule with the TA moderating, and you leave these two weeks with a prototype.
Weeks 14–15: Capstone symposium
Four sessions, three talks each. Fifteen minutes of talk, then ten of questions.
Objectives
- Students can identify a gap in a subject and defend it as genuine against the objection that it is closed or inconsequential.
- Students can convert a gap into a research question of tractable scope.
- Students can identify the smallest piece of evidence that would move a question forward.
- Students can execute a computational prototype and establish that its result is trustworthy.
- Students can present research, including what failed, and state what they would do next with reference to data that will exist.
Objectives carried through the whole term
These are not tied to a week. Every problem set, every board segment and every map is assessed against them.
- Students can formulate their own approach to a problem before consulting AI, and articulate what they did not know at the outset.
- Students can delegate a derivation or computation to AI and design a check capable of detecting an error in it.
- Students can distinguish a check that constrains a result from one that would pass whether or not the suspect term were present.
- Students can locate and diagnose an error in an AI-generated derivation and explain why it was not visible on first reading.
- Students can verify a symbolic or numerical computation by independent means and state what the verification does not cover.
- Students can reconstruct and defend a derivation at a blackboard under questioning.